Properties

Label 8-882e4-1.1-c2e4-0-4
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $333591.$
Root an. cond. $4.90232$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·4-s + 12·16-s − 112·23-s + 80·25-s − 112·29-s + 64·37-s − 128·43-s + 168·53-s + 32·64-s + 16·67-s − 448·71-s − 144·79-s − 448·92-s + 320·100-s − 112·107-s + 32·109-s + 56·113-s − 448·116-s − 468·121-s + 127-s + 131-s + 137-s + 139-s + 256·148-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 4-s + 3/4·16-s − 4.86·23-s + 16/5·25-s − 3.86·29-s + 1.72·37-s − 2.97·43-s + 3.16·53-s + 1/2·64-s + 0.238·67-s − 6.30·71-s − 1.82·79-s − 4.86·92-s + 16/5·100-s − 1.04·107-s + 0.293·109-s + 0.495·113-s − 3.86·116-s − 3.86·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 1.72·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(333591.\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3510406110\)
\(L(\frac12)\) \(\approx\) \(0.3510406110\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( ( 1 - p T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2:C_4$ \( 1 - 16 p T^{2} + 2752 T^{4} - 16 p^{5} T^{6} + p^{8} T^{8} \)
11$C_2^2$ \( ( 1 + 234 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2:C_4$ \( 1 - 320 T^{2} + 54400 T^{4} - 320 p^{4} T^{6} + p^{8} T^{8} \)
17$C_2^2:C_4$ \( 1 - 688 T^{2} + 960 p^{2} T^{4} - 688 p^{4} T^{6} + p^{8} T^{8} \)
19$C_2^2:C_4$ \( 1 - 900 T^{2} + 456870 T^{4} - 900 p^{4} T^{6} + p^{8} T^{8} \)
23$D_{4}$ \( ( 1 + 56 T + 1770 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 + 56 T + 2224 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$C_2^2:C_4$ \( 1 - 2276 T^{2} + 2834758 T^{4} - 2276 p^{4} T^{6} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 - 32 T + 2112 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$C_2^2:C_4$ \( 1 - 1040 T^{2} + 3520 p^{2} T^{4} - 1040 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 + 64 T + 3154 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$C_2^2:C_4$ \( 1 + 1564 T^{2} + 6450886 T^{4} + 1564 p^{4} T^{6} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 - 84 T + 3510 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$C_2^2:C_4$ \( 1 - 5316 T^{2} + 17444838 T^{4} - 5316 p^{4} T^{6} + p^{8} T^{8} \)
61$C_2^2:C_4$ \( 1 - 9152 T^{2} + 41434240 T^{4} - 9152 p^{4} T^{6} + p^{8} T^{8} \)
67$C_2$ \( ( 1 - 4 T + p^{2} T^{2} )^{4} \)
71$D_{4}$ \( ( 1 + 224 T + 22234 T^{2} + 224 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
73$C_2^2:C_4$ \( 1 - 12864 T^{2} + 88688448 T^{4} - 12864 p^{4} T^{6} + p^{8} T^{8} \)
79$D_{4}$ \( ( 1 + 72 T + 12210 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$C_2^2:C_4$ \( 1 - 3876 T^{2} + 69670758 T^{4} - 3876 p^{4} T^{6} + p^{8} T^{8} \)
89$C_2^2:C_4$ \( 1 - 6768 T^{2} + 136931136 T^{4} - 6768 p^{4} T^{6} + p^{8} T^{8} \)
97$C_2^2:C_4$ \( 1 - 15168 T^{2} + 120056640 T^{4} - 15168 p^{4} T^{6} + p^{8} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.13289161701323109438953795711, −7.05160239447109462749026149064, −6.56485683487337135954094419664, −6.21137757206913026942163617754, −6.09719077518429708349847565024, −6.03418757550348078163102814027, −5.86790485438770816818622477932, −5.38107257661742991266115208925, −5.30185849110404364488465430099, −5.14326559199692360055765040559, −4.66051227232911403095752325019, −4.32697635445282380217267207275, −4.03945372395814331351782668778, −3.90607536016086805434136293089, −3.87924743522667597957264045416, −3.41386930597029626589516281656, −2.83499444715074418581757376961, −2.80237158265824084842855153176, −2.68856687634626822887548036593, −2.09309190427223029507145427315, −1.71752323195233182485969362449, −1.65544269423089106510943434301, −1.44605420604289857169526031137, −0.63694385175640568915957479335, −0.095458874997617624182407994181, 0.095458874997617624182407994181, 0.63694385175640568915957479335, 1.44605420604289857169526031137, 1.65544269423089106510943434301, 1.71752323195233182485969362449, 2.09309190427223029507145427315, 2.68856687634626822887548036593, 2.80237158265824084842855153176, 2.83499444715074418581757376961, 3.41386930597029626589516281656, 3.87924743522667597957264045416, 3.90607536016086805434136293089, 4.03945372395814331351782668778, 4.32697635445282380217267207275, 4.66051227232911403095752325019, 5.14326559199692360055765040559, 5.30185849110404364488465430099, 5.38107257661742991266115208925, 5.86790485438770816818622477932, 6.03418757550348078163102814027, 6.09719077518429708349847565024, 6.21137757206913026942163617754, 6.56485683487337135954094419664, 7.05160239447109462749026149064, 7.13289161701323109438953795711

Graph of the $Z$-function along the critical line